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The second transpose of a derivation and weak amenability of the second dual Banach algebras. (English) Zbl 1354.46049

Summary: Let \(\mathcal A\) be a Banach algebra, \(\mathcal A^*\), \(\mathcal A^{**}\) and \(\mathcal A^{***}\) be its first, second and third dual, respectively. Let \(R:\mathcal A^{***} \to \mathcal A^*\) be the restriction map, \(J: \mathcal A^* \to \mathcal A^{***}\) be the canonical injection and \(\Lambda: \mathcal A^{***} \to \mathcal A^{***}\) be the composition of \(R\) and \(J\). Let \(D:\mathcal A \to \mathcal A^*\) be a continuous derivation and \(D^{\prime \prime}: \mathcal A^{**} \to \mathcal A^{***}\) be its second transpose. We obtain a necessary and sufficient condition for \(\Lambda \circ D^{\prime \prime}: \mathcal A^{**} \to (\mathcal A^{**})^*\) to be a derivation. We apply this to prove some results on weak amenability of second dual Banach algebras.

MSC:

46H20 Structure, classification of topological algebras
47B47 Commutators, derivations, elementary operators, etc.
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